Let \(\widetilde{T}_{\Omega ,A}\)  be defined as (REF ). By duality, for \(r\in (1,\,2]\) , it suffices to prove
\(\Big |\int _{\mathbb {R}^d}f_1(x)\widetilde{T}_{\Omega ,A}f_2(x)dx\Big |\lesssim r^{\prime 2}\sup _{\mathcal {S}}\mathcal {A}_{\mathcal {S},\,L^1,\,L^r}(f_1,\,f_2),\) 
holds for all bounded functions \(f_1\)  and \(f_2\)  with compact supports.
Let \(\mathbb {L}\subset \mathbb {R}^d\)  be a dyadic cube and \(\mathcal {Q}\)  be a stopping collection with top \(Q\) . We claim that
\(|\Lambda ^{A,s_Q,*}_Q(f_1\chi _{Q},f_2\chi _{3Q})|\le Cr^{\prime 2}|Q|\Vert f_1\Vert _{\mathcal {Y}_1(\mathcal {Q})}\Vert f_2\Vert _{\mathcal {Y}_r(\mathcal {Q})}+\sum _{\mathbb {L}\in \mathcal {Q},\,\mathbb {L}\subset Q}|\Lambda ^{A,s_\mathbb {L},*}_\mathbb {L}(f_1\chi _{\mathbb {L}},f_2\chi _{3\mathbb {L}})|.\) 
In fact, for fixed \(r\in (1,\,2]\) , \(f_1\) , \(f_2\)  with \({\rm supp}\,f_1\subset Q\) , \({\rm supp}\, f_2\subset 3Q\) , apply the Calderón-Zygmund decomposition on \(f_j\)  with respect to the family \(\mathcal {Q}\)  as described in Definition REF , and write
\(f_j=g_j+b_j,\,\, b_j=\sum _{\mathbb {L}\in \mathcal {Q}}b_{j,\,\mathbb {L}},\,,\,b_{j,\mathbb {L}}=\big (f_j-\langle f_j\rangle _{\mathbb {L}}\big )\chi _{\mathbb {L}}(x).\) 
Then we have
\(\Lambda ^{A,s_Q,*}_Q(f_1,f_2)=\Lambda ^{A,*}_{\mathcal {Q}}(g_1,f_2)+\Lambda ^{A,*}_{\mathcal {Q}}(b_1,f_2)+\sum _{\mathbb {L}\in \mathcal {Q},\,\mathbb {L}\subset Q}\Lambda ^{A,s_\mathbb {L},*}_\mathbb {L}(f_1\chi _{\mathbb {L}},f_2\chi _{3\mathbb {L}}).\) 
Inequality (REF ) tells us that
\(|\Lambda ^{A,*}_{\mathcal {Q}}(g_1,f_2)|\lesssim r^{\prime 2}\Vert g_1\Vert _{L^{r^{\prime }}(\mathbb {R}^d)}\Vert f_2\Vert _{L^r(\mathbb {R}^d)}\lesssim r^{\prime 2}|Q|\Vert f_1\Vert _{\mathcal {Y}_1(\mathcal {Q})}\Vert f_2\Vert _{\mathcal {Y}_r(\mathcal {Q})}.\) 
On the other hand, Lemma REF  indicates that
\(|\Lambda ^{A,*}_{\mathcal {Q}}(b_1,f_2)|\lesssim r^{\prime 2}\Vert b_1\Vert _{\dot{\mathcal {X}}_1(\mathcal {Q})}\Vert f_2\Vert _{\mathcal {Y}_r(\mathcal {Q})}\lesssim r^{\prime 2}|Q|\Vert f_1\Vert _{\mathcal {Y}_1(\mathcal {Q})}\Vert f_2\Vert _{\mathcal {Y}_r(\mathcal {Q})}.\) 
Combining these two estimates may leads to (REF ).
With (REF ) in hand, by repeating the proof of Theorem C in [1]}, we obtain that for each bounded function \(f_1\)  and \(f_2\) , there exists a sparse family of cubes \(\mathcal {S}\) , such that
\(\Big |\int _{\mathbb {R}^d}f_1(x)\widetilde{T}_{\Omega ,\,A}f_2(x)dx\Big |\lesssim r^{\prime 2}\mathcal {A}_{\mathcal {S},\,L^1,\,L^r}(f_1,\,f_2).\) 
This completes the proof of Theorem REF .
